The influence of a presence of a heavy atom on the spin-spin coupling constants between two light nuclei in organometallic compounds and halogen derivatives Artur Wodyński and Magdalena Pecul Citation: The Journal of Chemical Physics 140, 024319 (2014); doi: 10.1063/1.4858466 View online: http://dx.doi.org/10.1063/1.4858466 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nuclear spin–spin coupling constants from regular approximate relativistic density functional calculations. I. Formalism and scalar relativistic results for heavy metal compounds J. Chem. Phys. 113, 936 (2000); 10.1063/1.481874 Spin–spin coupling constants between carbon13 and bromine in bromomethanes J. Chem. Phys. 67, 3803 (1977); 10.1063/1.435322 Signs of Spin—Spin Coupling Constants between Methyl Protons and Ring Fluorine Nuclei in Fluorotoluene Derivatives. Further Evidence for a Positive Hyperfine Interaction in the C–F Bond J. Chem. Phys. 47, 5037 (1967); 10.1063/1.1701756 LongRange Spin—Spin Interaction between Nuclei in the Saturated Compounds J. Chem. Phys. 41, 315 (1964); 10.1063/1.1725869 Nuclear Spin—Spin Coupling Involving Heavy Nuclei. The Coupling between Hg199 and H1 Nuclei in CH3HgX and CH3CH2HgX Compounds J. Chem. Phys. 39, 1330 (1963); 10.1063/1.1734435
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THE JOURNAL OF CHEMICAL PHYSICS 140, 024319 (2014)
The influence of a presence of a heavy atom on the spin-spin coupling constants between two light nuclei in organometallic compounds and halogen derivatives ´ Artur Wodynski and Magdalena Pecula) Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warszawa, Poland
(Received 6 October 2013; accepted 13 December 2013; published online 14 January 2014) The 1 JCC and 1 JCH spin-spin coupling constants have been calculated by means of density functional theory (DFT) for a set of derivatives of aliphatic hydrocarbons substituted with I, At, Cd, and Hg in order to evaluate the substituent and relativistic effects for these properties. The main goal was to estimate HALA (heavy-atom-on-light-atom) effects on spin-spin coupling constants and to explore the factors which may influence the HALA effect on these properties, including the nature of the heavy atom substituent and carbon hybridization. The methods applied range, in order of reduced complexity, from Dirac-Kohn-Sham method (density functional theory with four-component Dirac-Coulomb Hamiltonian), through DFT with two- and one-component Zeroth Order Regular Approximation (ZORA) Hamiltonians, to scalar non-relativistic effective core potentials with the non-relativistic Hamiltonian. Thus, we are able to compare the performance of ZORA-DFT and Dirac-Kohn-Sham methods for modelling of the HALA effects on the spin-spin coupling constants. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4858466] I. INTRODUCTION
In the last two decades great progress has been made in methodology of relativistic quantum chemical calculations and in development of computer codes for this purpose. The importance of the relativistic effects is getting widely recognized in the scientific community and more and more molecular properties are being calculated using relativistic Hamiltonians. Developments in perturbational density functional theory (DFT) allowed for simultaneous inclusion of the relativistic and electron correlation effects for a variety of molecular properties for fairly large systems. Four- and twocomponents Hamiltonians are applied within the framework of density functional theory for calculations of such properties as linear molecular polarizabilities,1–5 non-linear molecular polarizabilities and other high-order optical properties,6–8 nuclear shielding constants,9–15 and indirect spin-spin coupling constants.16–20 The nuclear shielding constants and the spinspin coupling constants of heavy nuclei are of special interest in this respect, since they depend on electron density in the vicinity of the nucleus, and it has been recognized early on21 that electron velocities (and consequently the relativistic effects) are the largest in this region. The presence of a heavy nucleus in a molecule affects not only the Nuclear Magnetic Resonance (NMR) properties of the heavy nucleus in question, but influences also the shielding constants and indirect spin-spin coupling constants of the nearby light nuclei. This phenomenon is called, after Pyykkö et al.,22 the heavy-atom-on-light-atom (HALA) effect. While the relativistic effects on the shielding constants (or related to them chemical shifts) of light nuclei neighbouring heavy a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2014/140(2)/024319/8/$30.00
atoms are relatively well investigated in the literature23–30 (although mainly for proton and carbon shielding constants in halogen derivatives, much less is known about metalorganic compounds), the parallel phenomenon occurring for the nuclear spin-spin coupling constants is almost unexplored. There is a handful of papers dealing with the situation when a heavy atom mediates the geminal coupling between two light nuclei,31–35 and there seems to be a consensus that in this case the total relativistic effect is usually dominated by the scalar effects (unlike the HALA effect on the shielding constants, for which spin-orbit coupling plays a crucial role). There are, however, very few investigations concerning the situation where the heavy atom is not in the coupling path. One of those is our study on heavy metal cyanides,29 where we have shown that the relativistic effect on the 1 JCN spin-spin coupling constant may exceed 20% (for mercury cyanide) or even 40% (for gold cyanide) of the total value of the coupling. These findings inspired us to look more closely at the influence of the presence of heavy atom on the one-bond spin-spin coupling constant of the nearby light nuclei, and to investigate what factors determine the magnitude of the effect and whether the dominant role of the scalar relativistic effects is a general rule. The systems under study are derivatives of aliphatic hydrocarbons substituted with I, At, Cd, and Hg. This choice allowed us to study both one-bond carbon-carbon and onebond proton-carbon coupling constants and to explore the factors which may influence the HALA effect on these properties: the nature of the heavy atom substituent and carbon hybridization. These factors have been found important for carbon chemical shifts.30, 36 The calculations are carried out using density functional theory with the zeroth-order regular approximation (ZORA) Hamiltonian (with the spin-orbit term included), as implemented by Autschbach for the spin-spin
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coupling constants,16, 17 with the four-component DiracCoulomb Hamiltonian (as implemented recently by Saue),37 and with the non-relativistic Hamiltonian combined with scalar effective core potentials (ECPs), since our secondary aim is to compare the performance of DFT with scalar ECPs, ZORA-DFT, and Dirac-Kohn-Sham (DKS) methods for modelling of the HALA effects on the spin-spin coupling constants. The paper is organized as follows. After the description of the computational methods, we discuss the calculated 1 JCC and 1 JCH coupling constants, for each considering first the influence of the heavy atom charge and carbon hybridization on the HALA effect, and then moving to the methodological issues, comparing the results of calculations with ZORA, Dirac-Coulomb, and Schrödinger Hamiltonians, the latter both with relativistic ECPs and all-electron basis set. Finally, the results are summarized and main conclusions are presented. II. COMPUTATIONAL DETAILS A. Geometry optimization
The geometric parameters of the isolated molecules under study have been obtained by means of geometry optimization carried out using DFT with the zeroth-order regular approximation Hamiltonian38 (the spin-orbit coupling term included) as implemented in the ADF39 program with VWN40 +Becke8841 and Perdew8642 exchange-correlation functional (the functional is denoted as BP86 in ADF, but actually differs from the functional usually indicated by this acronym by using VWN instead of the local correlation PZ81 functional43 ) and the TZ2P basis set. The same geometry parameters (obtained by means of spin-orbit ZORA calculation) have been used for all (DKS, ZORA, ECP) calculations of the spin-spin coupling constants in order to separate the effects of a different computational model on the molecular geometry and the spin-spin coupling constants. B. DKS calculations of the spin-spin coupling constants
The four-component DKS calculations of the spin-spin coupling constants have been carried out with a local version of the DIRAC program,44 including the recent developments by Saue.37 The large and small components of the wave function have been connected by unrestricted kinetic balance, as implemented in DIRAC.44 The Perdew, Burke, and Ernzerhof functional with Adamo and Barone’s HF exchange contribution (PBE0)45 has been used together with the uncontracted triple-ζ basis set with additional tight functions (aug-cc-pVTZ-J46 ) for carbon and hydrogen, and uncontracted triple-ζ Dyall’s basis set (dyall.v3z47 ) for mercury, cadmium, iodine, and astatine. All calculations have been performed with the Gaussian charge distribution model, as default in DIRAC. C. ZORA calculations of the spin-spin coupling constants
The ZORA results have been obtained using the DFT as implemented in the ADF program, employing the PBE0
J. Chem. Phys. 140, 024319 (2014)
hybrid functional. (First-order potential of the hybrid PBE0 functional have been used during the calculations of the spinspin coupling constants.) The results obtained with both onecomponent scalar ZORA Hamiltonian (denoted as sc-ZORA) and with two-component spin-orbit ZORA Hamiltonian (denoted so-ZORA) will be presented. We have used triple-ζ Slater basis set with additional tight functions (jcpl48 ) for hydrogen, carbon, mercury, and iodine and the TZ2P (tripleζ +2 polarization functions basis set) for cadmium and astatine. All calculations have been performed with the Gaussian charge distribution model. D. ECP calculations of the spin-spin coupling constants
The effective core potential results have been obtained using the Gaussian 0949 program. We have employed the PBE0 functional, the aug-cc-pVTZ-J basis set for carbon and hydrogen, and several effective core potentials for the heavy elements: LANL2DZ for iodine,50 cadmium,51 and mercury,51 MWB28 for cadmium,52 MWB46 for cadmium53 and iodine,54 MWB60 for mercury,52 MWB78 for astatine55 and mercury,55 MDF28 for cadmium56 and iodine,57 MDF46 for iodine,58 MDF60 for astatine57 and mercury,59 and MDF78 for astatine.58 E. Non-relativistic calculation of the spin-spin coupling constants
Two sets of non-relativistic calculations of the spin-spin coupling constants have been performed. The first set has been carried out using DFT as implemented in the ADF program with the PBE0 functional. We have used the tripleζ Slater jcpl basis set available in the ADF program for hydrogen, carbon, mercury, and iodine and TZ2P for cadmium and astatine. The second set of calculations has been performed using DFT as implemented in Dalton 201160 program. The PBE0 functional, the Gauss-type uncontracted aug-cc-pVTZ-J basis set for carbon and hydrogen, and the uncontracted dyall.v3z basis set47 for mercury, cadmium, iodine, and astatine have been used. This allows us to estimate the effect of using different basis sets in ADF and DIRAC calculations. III. RESULTS AND DISCUSSION
Below, we are going to discuss the influence of the presence of a heavy atom on the spin-spin coupling constants between two light nuclei (first 1 JCC , then 1 JCH ). For each coupling constant, we first consider the influence of various factors (the charge of the heavy nucleus, the carbon hybridization, the relative magnitude of the scalar and spin-orbit terms), and then compare the ZORA-DFT and DKS results and discuss the performance of the ECPs in rendering the scalar relativistic effects (Secs. III A 2 and III B 2, respectively). All results discussed in Subsections III A 1 and III B 1 have been obtained with the ADF package using the ZORA or non-relativistic Hamiltonians. Additional calculations, reported in Secs. III A 2 and III B 2, have been performed with
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TABLE I. The 1 JCC coupling constants calculated with the PBE0 functional using different Hamiltonians. Jcpl basis sets (TZ2P for cadmium and astatine) have been used for nonrelativistic, sc-ZORA, and so-ZORA calculations, aug-cc-pVTZ-J basis set for carbon and hydrogen, and dyall.v3z basis set for cadmium, mercury, iodine, astatine for DKS calculations. The relativistic term has been calculated as difference between so-ZORA and nonrelativistic results. Nonrelativistic (Hz)
sc-ZORA (Hz)
so-ZORA (Hz)
Relativistic term (Hz)
DKS (Hz)
Substituent effect (Hz)
ICCH ICHCH2 ICH2 CH3 AtCCH AtCHCH2 AtCH2 CH3 CH3 CdCCH CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCCH CH3 HgCHCH2 CH3 HgCH2 CH3
208.9 80.2 34.0 199.3 80.0 34.6 125.2 59.2 31.0 126.0 59.1 30.6
207.7 79.7 33.6 193.5 78.9 34.0 130.2 60.8 31.4 144.1 64.9 32.1
206.3 80.1 34.1 176.2 78.9 35.6 130.3 60.7 31.3 146.0 64.9 32.0
− 2.6 − 0.1 0.1 − 23.1 − 1.1 1.0 5.1 1.5 0.3 20.0 5.9 1.4
206.8 79.7 34.6 172.7 78.2 36.2 130.2 60.6 30.8 144.7 64.7 32.3
9.0 8.1 1.8 − 21.1 6.8 3.3 − 67.0 − 11.4 − 1.0 − 51.2 − 7.1 − 0.3
HCCH CH2 CH2 CH3 CH3
197.2 72.2 32.4
197.3 72.1 32.3
197.3 72.0 32.3
... ... ...
... ... ...
... ... ...
the Dirac-Coulomb Hamiltonian and Schrödinger Hamiltonian combined with scalar relativistic ECPs. A. 1 JCC coupling constants
The 1 JCC coupling constants calculated using the Schrödinger, ZORA and Dirac-Coulomb Hamiltonians are shown in Table I. Table I contains also the “relativistic effect,” calculated as a difference between the so-ZORA and non-relativistic results, and the “substituent effect,” calculated as a difference between the so-ZORA result for the compound under study and the corresponding unsubstituted aliphatic hydrocarbon. 1. General considerations a. Influence of the charge of the heavy nucleus. The relativistic effects for the 1 JCC spin-spin coupling constants under study are sizable for substituents from the 6th row of periodic table (for mercury and astatine compounds) in comparison with the total substituent effects, but nearly negligible (10% or less) for the 5th row compounds. In the case of mercury compounds, taking into account the relativistic effects significantly lower the calculated substituent effects. The same can be observed for cadmium derivatives, but in this case the relativistic effect is only about 10% of the total substituent effect. For iodine derivatives the relativistic effect on 1 JCC is negligible (of the order of magnitude of numerical accuracy), with the exception of ICCH, where it lowers the calculated substituent effect by about 30%. In the case of AtCCH, the total substituent effect is actually dominated by the relativistic effect (the non-relativistic calculations lead to a very similar result as for HCCH). For other astatine derivatives, the relativistic effect is smaller, but still quite significant in comparison with the substituent effect. (Of the same sign for AtCH2 CH3 , of the opposite sign for AtCHCH2 .) The magnitude of the relativistic effect changes strongly with carbon hybridization (as discussed in more detail below),
but the trends in these changes are very similar in both series of structural analogs. The ratio between the relativistic effects for astatine derivative and its iodine analog is always about 11, and about 4 for mercury derivative and its cadmium analog. b. Influence of the carbon hybridization. The magnitude of the substituent and relativistic effects on the 1 JCC spin-spin coupling constants depends strongly on the hybridization of carbon atoms. The largest effects are observed for the systems with the sp hybridization and the smallest (at least one order of magnitude smaller) effects for the systems with the sp3 hybridization (except for halides, where the relativistic effects for the sp2 and sp3 hybridization are of comparable magnitude). It is however worth noting that for the compounds containing the 12th group elements the relative contribution of the relativistic effect to the substituent effect actually increases when going from the sp to sp3 hybridization, since the substituent effect decreases to a larger extent than the relativistic effect. c. Influence of the spin-orbit coupling. The ratio of the scalar and spin-orbit contributions to the total relativistic effect on the 1 JCC spin-spin coupling constant depends strongly on the nature of the heavy atom substituent. For halides both contributions are, as a rule, of similar magnitude, except for AtCHCH2 , where the scalar term dominates. Very small values of the relativistic effect on 1 JCC in ICHCH2 and ICH2 CH3 result from mutual cancelling of the scalar and spin-orbit term, each about 0.5 Hz, but with the opposite signs. In mercury and cadmium derivatives, nearly the total relativistic effect comes from the scalar term, since the spin-orbit term contributes less than 10% to it (except for CH3 CdCH2 CH3 , but here the relativistic effect is very small). Larger role of spin-orbit coupling in the molecular properties of halides seems a general rule, connected with p-character of the heavy atom.
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TABLE II. Comparison of the individual terms of 1 JCC in AtCCH and CH3 HgCCH, obtained using nonrelativistic, sc-ZORA, and so-ZORA Hamiltonians. AtCCH (Hz)
CH3 HgCCH (Hz)
Nonrelativistic (ADF) DSO PSO FC SD
199.3 0.2 10.4 176.2 12.4
126.1 0.3 4.4 111.3 10.1
sc-ZORA DSO PSO FC SD
193.5 0.2 11.1 169.7 12.5
144.2 0.3 5.8 127.5 10.6
so-ZORA DSO PSO FC SD FC+SD/PSO cross term
176.2 0.2 10.3 152.8 10.5 2.4
146.0 0.3 5.9 129.4 10.8 − 0.3
Inspired by the work of Autschbach et al.,17 we decided to examine the influence of the spin-orbit coupling on the individual terms of the spin-spin coupling constant. In the twocomponent computation, there exists a cross term between the spin-dependent Fermi contact (FC) and spin-dipole (SD) terms, and the paramagnetic spin-orbital (PSO) term, denoted later as FC+SD/PSO. In most cases the spin-orbit contribution to spin-spin coupling is dominated by this term. We have therefore compared (Table II) the individual terms of 1 JCC in AtCCH (where the spin-orbit contribution is the largest) and in CH3 HgCCH, where the spin-orbit contribution is negligible with respect to the scalar term. The results show that in the case of CH3 HgCCH the FC+SD/PSO cross term is about 17% of the spin-orbit contribution (about −0.3 Hz), and approximately 15% of the spin-orbit contribution (about 2.4 Hz) in AtCCH. Therefore, unlike for the couplings of heavy nuclei analyzed by Autschbach et al.,17 the FC+SD/PSO cross term is sizeable for 1 JCC , but does not dominate the spin-orbit contribution. Inclusion of the spin-orbit coupling changes predominantly the pure FC term. 2. Comparison of different computational methods
In this paragraph we compare the one-component and two-component ZORA results with the results of other available computational methods including the relativistic effects: four-component all-electron Dirac-Kohn-Sham method and effective core potentials. It should be stressed that comparison of ZORA and Dirac-Kohn-Sham results or ZORA and ECP results is not straightforward, since, out of necessity, different basis sets are used (ADF employs Slater orbital basis set, while in DIRAC or Gaussian program Gauss orbitals are used). For that reason we have also performed comparison of the results obtained using the Slater and Gauss orbitals with the non-relativistic Hamiltonians. Very good agreement between the results obtained with the jcpl (Slater type) basis set and the aug-cc-pVTZ-J/dyall.v3z (Gauss type) has been ob-
served (see Ref. 61 for the data in Table 1). In most cases the results differ less than 0.5 Hz (for several cases the difference is between 0.5 and 1.0 Hz). Only for AtCCH a significantly larger difference is observed (about 2.8 Hz). Considering this, the use of different basis sets should not influence materially the comparison between the ZORA-DFT and DKS results, except when the relativistic effect is very small. a. Comparison between so-ZORA and DKS. Inspection of the results in Table I leads to the conclusion that so-ZORA reproduces the DKS results very well. Explicit inclusion of small spinor during the calculation does not change significantly the calculated 1 JCC couplings in comparison with twocomponent ZORA approximation: in most cases the differences are less than 1 Hz, and seems to be caused mostly by the change of the basis set (see above) rather than the picture change effects. Only for the systems with the heaviest metallic substituents and sp-hybridized carbon nuclei (i.e., CH3 HgCCH and AtCCH) the ZORA-DFT—DKS difference is slightly bigger (e.g., 3.4 Hz in the case of AtCCH) but a significant part of this change can be again attributed to the basis set effect (about 2.8 Hz, see above). b. Comparison between sc-ZORA and ECPs methods. The JCC spin-spin coupling constants calculated using scalar ECPs are compared in Table III with the one-component scalar-only ZORA results. Again, it should be pointed out that a different basis set has been used for sc-ZORA and ECP calculations. Two types of ECPs have been investigated: Los Alamos ECPs (LANL2DZ) and series of Stuttgart ECPs (MDF and MWB with small and big core replacement). LANL2DZ does not reproduce the sc-ZORA results very well, and actually it does not offer any consistent improvement over the non-relativistic calculations. The same is true of the large-core MDF and MWB pseudopotentials, so it seems to be mainly a matter of replacing too many core electrons by the effective potential. The small-core MWB results are in excellent agreement with sc-ZORA. A slightly worse agreement is observed for small-core MDF results, but they are still a big improvement over the non-relativistic calculations, and the differences between the MWB60 and MDF60 results are usually very small. One of a very few cases when a sizeable discrepancy (about 4 Hz) between the MWB60 and MDF60 results is observed is 1 JCC in CH3 HgCCH. We have performed additional calculations with a more recent version of MDF6056 (denoted in Table III as MDF60(new)) and obtained results much closer to MWB60; thus the observed difference is most likely a consequence of different fitting procedures employed to obtain the two ECPs.
1
B. 1 JCH coupling constants 1
JCH coupling constants in the systems under study are of two types, as far as the placement of the coupled nuclei with respect to the heavy atom is concerned. First we are going to discuss the 1 JCH coupling constants of the carbon nuclei directly bound with the heavy atom (1 JCα H ), and later the 1 JCH coupling constants involving the carbon nuclei in beta
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TABLE III. Comparison of 1 JCC calculated with different types of effective core potentials on heavy atoms and aug-cc-pVTZ-J basis set on C and H, using PBE0 functional.
ICCH ICHCH2 ICH2 CH3 AtCCH AtCHCH2 AtCH2 CH3 CH3 CdCCH CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCCH CH3 HgCHCH2 CH3 HgCH2 CH3
LANL2DZ (Hz)
MDF Largea (Hz)
MDF Smallb (Hz)
MDF60 Newc (Hz)
MWB Largea (Hz)
MWB Smallb (Hz)
sc-ZORA (Hz)
210.0 79.5 33.5 ... ... ... 121.3 58.0 30.7 128.0 60.8 31.1
209.5 79.4 33.5 197.9 78.9 33.9 ... ... ... ... ... ...
207.6 79.3 33.6 194.3 78.6 34.0 131.4 60.7 31.4 140.6 63.1 31.8
... ... ... ... ... ... ... ... ... 144.1 64.5 32.1
210.7 79.5 33.5 196.5 78.7 33.9 120.4 151.8 30.7 130.6 60.9 31.3
... ... ... ... ... ... 131.5 60.8 31.5 144.3 64.6 32.1
207.7 79.7 33.6 193.5 78.9 34.0 130.2 60.8 31.4 144.2 64.9 32.2
a
Large-core pseudopotential. Small-core pseudopotential. c Small-core pseudopotential (60 core electrons replaced by a pseudopotential in Hg) reparametrized.56 b
position with respect to the heavy atom. In the latter case, for the system of the RCHCH2 type, the coupled proton can be either in cis or in trans position with respect to the heavy atom, and for the RCH2 CH3 systems the dihedral angle between the heavy atom–alpha carbon and hydrogen–beta carbon may play a role, so these factors will be discussed. The computational results are presented in Table IV. 1. General considerations a. Influence of the charge of the heavy nucleus. The relativistic effects on the 1 JCH coupling constants in iodinesubstituted hydrocarbons are 1 Hz or less (even when carbon is in the α position with respect to the heavy atom). Somewhat larger effects (but still rather smaller than those resulting from other changes in computational protocol, e.g., the use of a different basis set) can be observed for astatinesubstituted analogs. The only case when the relativistic effect on 1 JCH constitutes a majority of the substituent effect (apart from CH3 CdCH2 CH3 , where the substituent effect is very small) takes place in organomercury compounds—there the relativistic effect can even exceed 10 Hz. The ratio between the relativistic effect for 1 JCα H in organomercury compound and its cadmium analog is about 3 (similarly as for 1 JCC ), and a similar regularity can be observed for the 1 JCβ H couplings (but not for iodine/astatine analogs). Not surprisingly, the relativistic effect weakens with the distance of the coupled nuclei from the heavy atom: it is much larger for 1 JCα H than for 1 JCβ H . In the case of 1 JCβ H for cadmium compound with sp2 and sp3 carbon hybridization the relativistic contributions do not exceed 1.4 Hz, and for iodine compounds they are even smaller (about 0.4 Hz). The biggest contribution for 1 JCβ H coupling constants is observed for CH3 HgCCH, where the relativistic term is 10.0 Hz. Interestingly, the relativistic calculations lead in this case to much smaller substituent effect than the non-relativistic ones. b. Influence of the carbon hybridization and molecular geometry. Similarly as for 1 JCC , carbon hybridization influences
the magnitude of the relativistic effects on 1 JCα H . They are bigger for the systems with the sp2 hybridization than for the systems with the sp3 hybridization (compare 15.3 Hz for CH3 HgCHCH2 and 6.6 Hz for CH3 HgCH2 CH3 ). Only for iodine derivatives a reverse trend is observed but in this case the relativistic terms are small (less than 0.7% of the total spinspin coupling constant) and therefore sensitive to numerical noise. In the case of 12th group of periodic table and astatine compounds the ratio between the relativistic effects for the analogous systems with sp2 and sp3 hybridization is about 2–2.5. The relativistic effects on 1 JCβ H are visibly affected by the heavy atom–carbon–carbon-hydrogen dihedral angle. When cis and trans configurations of cadmium and mercury derivatives are compared, always the coupling of the proton in the trans position is more affected by the relativistic effects (and exhibits larger variation with carbon hybridization). This is not necessarily true for iodine and astatine derivatives, but, as mentioned above, the relativistic effects are usually negligibly small there. c. Influence of the spin-orbit coupling. A comparison of the so-ZORA and sc-ZORA results shows that the spin-orbit coupling effect is as a rule negligible in comparison with the scalar term for all types of 1 JCH , except for astatine derivatives. For 1 JCα H in AtCHCH2 the spin-orbit coupling term is about −4.9 Hz (the scalar term is only 0.4 Hz), while for 1 JCβ H in AtCCH it is about −4.7 Hz (to be compared with the scalar term of 0.2 Hz) and for 1 JCβ Htrans in AtCHCH2 it is about −2.0 Hz (the scalar term 0.1 Hz). It is noticeable that for astatine derivatives with sp3 hybridization the spin-orbit coupling term is much smaller than for derivatives with sp2 and sp carbon.
2. Comparison of different computational approaches a. Comparison between so-ZORA and DKS. Inspection of the results (see Table IV) for all 1 JCH confirms that the two-component ZORA approximation reproduces the
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TABLE IV. 1 JCH calculated with PBE0 functional at different levels of theory. The Jcpl basis sets (TZ2P for cadmium and astatine) have been used for nonrelativistic, sc-ZORA, and so-ZORA calculations, aug-cc-pVTZ-J basis set for carbon and hydrogen, and dyall.v3z basis set for cadmium, mercury, iodine, astatine for DKS calculations. The relativistic term has been calculated as a difference between so-ZORA and nonrelativistic results.
1J Cα H
1J Cβ Hcis
1J Cβ Htrans
1J Cβ H
1J Cα H 1J CH
–CH3
ICHCH2 ICH2 CH3 AtCHCH2 AtCH2 CH3 CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCHCH2 CH3 HgCH2 CH3 ICHCH2 ICH2 CH3 AtCHCH2 AtCH2 CH3 CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCHCH2 CH3 HgCH2 CH3 ICHCH2 ICH2 CH3 AtCHCH2 AtCH2 CH3 CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCHCH2 CH3 HgCH2 CH3 ICCH AtCCH CH3 CdCCH CH3 HgCCH CH3 CdCCH CH3 HgCCH CHCH CH2 CH2 CH3 CH3 CH4
Nonrelativistic (Hz)
sc-ZORA (Hz)
so-ZORA (Hz)
Relativistic term (Hz)
DKS (Hz)
Substituent effect (Hz)
196.1 151.9 197.0 153.8 139.0 126.0 137.5 125.0 167.9 131.6 167.7 131.6 159.0 124.7 160.9 125.3 161.7 122.2 162.1 122.4 153.3 124.1 151.2 122.9 270.7 269.6 239.7 237.4 129.7 128.1 262.2 159.1 125.2 125.7
197.2 152.5 197.4 154.5 143.8 128.3 152.1 131.6 167.7 131.5 167.2 131.2 159.1 125.0 160.3 125.8 162.4 122.6 162.2 122.4 154.6 124.5 155.8 124.4 271.6 269.4 242.6 246.8 132.1 135.0 262.4 159.2 125.3 125.8
197.0 153.0 192.5 155.2 143.7 128.2 152.8 131.6 167.6 131.4 166.8 130.7 159.1 124.9 160.3 125.8 161.9 122.6 160.2 122.4 154.6 124.5 156.1 124.3 271.1 264.7 242.6 247.4 132.0 134.9 262.4 159.1 125.2 125.8
0.9 1.1 − 4.5 1.3 4.7 2.2 15.3 6.6 − 0.3 − 0.2 − 0.9 − 0.9 0.0 0.2 − 0.6 0.5 0.2 0.4 − 1.9 0.0 1.4 0.4 4.9 1.5 0.4 − 4.9 2.9 10.0 2.3 6.8 ... ... ... ...
195.9 152.9 190.9 155.0 143.4 127.8 151.9 131.0 166.8 131.1 166.0 130.3 158.2 124.7 159.3 125.4 160.9 122.2 159.1 121.9 153.9 124.3 155.7 124.4 270.1 263.7 241.6 247.1 131.3 133.9 ... ... ... ...
37.9 27.8 33.4 30.0 − 15.4 2.9 − 6.3 6.4 8.5 6.2 7.7 5.4 − 0.1 − 0.3 1.2 0.6 2.8 − 2.6 1.0 − 2.8 − 4.5 − 0.8 − 3.0 − 0.9 8.7 2.3 − 19.8 − 14.9 6.2 9.1 ... ... ... ...
four-component DKS results very well, especially when the fact that two different basis sets (Slater- and Gauss-type) are used is taken into account. The biggest difference between soZORA/DFT and DKS (about 1.6 Hz) is observed for 1 JCα H in AtCHCH2 , but it seems to originate from different basis sets used (compare the non-relativistic results obtained with the Slater- and Gauss-type basis sets tabulated in Table 2 in the supplementary material61 ). For the other 1 JCH spin-spin coupling constants the differences between so-ZORA/DFT and DKS are in most cases less than 1.0 Hz and also seem to come mostly from using different basis sets. Thus, we conclude that so-ZORA/DFT is very efficient in rendering HALA effects on the 1 JCH coupling constants. In fact, in most cases (apart from astatine derivatives), using one-component scalar ZORA is sufficient.
b. Comparison between sc-ZORA and ECPs methods. The JCH coupling constants calculated using different scalar relativistic effective core potentials are compared with the sc-
1
ZORA results in Table 3 in the supplementary material.61 The observations are similar as for the 1 JCC coupling constants. ECPs in general seem suitable for the purpose, and the best results are obtained when small-core ECPs are employed. Among small-core ECPs, MWB appears to perform slightly better than MDF, but the differences are small.
IV. SUMMARY AND CONCLUSIONS
The 1 JCC and 1 JCH spin-spin coupling constants have been calculated by means of density functional theory for a set of derivatives of aliphatic hydrocarbons substituted with I, At, Cd, and Hg in order to evaluate the substituent and relativistic effects for these properties. The methods applied range, in order of reduced complexity, from Dirac-KohnSham method (density functional theory with four-component Dirac-Coulomb Hamiltonian), through DFT with two- and one-component zeroth order regular approximation Hamiltonians, to scalar non-relativistic effective core potentials with
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the non-relativistic Hamiltonian. The most important observations can be summarized as follows. The relativistic effects on the spin-spin coupling constants under study are at most about 25 Hz (for 1 JCC in organomercury compounds), and constitute up to 30% of the substituent effect (defined as a difference between the value of the coupling in the compound under study and the corresponding hydrocarbon). Predictably, they vary with the charge of the heavy nucleus. For 1 JCC , the ratio between relativistic effects for astatine derivative and its iodine analog is always about 11, and for mercury derivative and its cadmium analog about 4. The ratio between the relativistic effect for 1 JCα H in organomercury compound and its cadmium analog is about 3 and a similar regularity can be observed for the 1 JCβ H couplings (but not for the iodine/astatine analogs). The magnitude of the relativistic effects on the 1 JCC and 1 JCH coupling constants depends very much on carbon hybridization. For both couplings the largest effects are observed for the sp hybridization. As expected, the relativistic effects are larger for 1 JCα H than for 1 JCβ H , although even for the latter they can be non-negligible (10 Hz for CH3 HgCCH). 1 JCβ H are larger when the proton is in the trans position to the heavy substituent than when it is in a cis position. A comparison of the results obtained by means of different methods of including the relativistic effects indicates that ZORA-DFT (with the spin-orbit term included) reproduces the DKS results very well. The scalar contributions dominate the total relativistic effect on 1 JCC and 1 JCH in mercury and cadmium derivatives, while for 1 JCC and 1 JCH in halides scalar and spin-orbit effects contribute to a similar degree. The performance of scalar ECPs depends, obviously, on the relative weight of the scalar and relativistic effects, but when the former are dominant, ECPs reproduce correctly the results obtained with more elaborate relativistic method, provided the outer core electrons on the heavy atom are accounted for explicitly (“small core” types of ECPs).
ACKNOWLEDGMENTS
This work has received support from the Polish National Science Centre via the Grant No. N N204 148565, and from the Wrocław Centre for Networking and Supercomputing through a grant of computer time. MPD/2010/4 project, realized within the MPD programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, is acknowledged for a fellowship to A. Wody´nski. The project has been carried out with the use of CePT infrastructure financed by the European Union - the European Regional Development Fund within the Operational Programme “Innovative economy” for 2007-2013. 1 P.
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THE JOURNAL OF CHEMICAL PHYSICS 140, 024319 (2014)
The influence of a presence of a heavy atom on the spin-spin coupling constants between two light nuclei in organometallic compounds and halogen derivatives ´ Artur Wodynski and Magdalena Pecula) Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warszawa, Poland
(Received 6 October 2013; accepted 13 December 2013; published online 14 January 2014) The 1 JCC and 1 JCH spin-spin coupling constants have been calculated by means of density functional theory (DFT) for a set of derivatives of aliphatic hydrocarbons substituted with I, At, Cd, and Hg in order to evaluate the substituent and relativistic effects for these properties. The main goal was to estimate HALA (heavy-atom-on-light-atom) effects on spin-spin coupling constants and to explore the factors which may influence the HALA effect on these properties, including the nature of the heavy atom substituent and carbon hybridization. The methods applied range, in order of reduced complexity, from Dirac-Kohn-Sham method (density functional theory with four-component Dirac-Coulomb Hamiltonian), through DFT with two- and one-component Zeroth Order Regular Approximation (ZORA) Hamiltonians, to scalar non-relativistic effective core potentials with the non-relativistic Hamiltonian. Thus, we are able to compare the performance of ZORA-DFT and Dirac-Kohn-Sham methods for modelling of the HALA effects on the spin-spin coupling constants. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4858466] I. INTRODUCTION
In the last two decades great progress has been made in methodology of relativistic quantum chemical calculations and in development of computer codes for this purpose. The importance of the relativistic effects is getting widely recognized in the scientific community and more and more molecular properties are being calculated using relativistic Hamiltonians. Developments in perturbational density functional theory (DFT) allowed for simultaneous inclusion of the relativistic and electron correlation effects for a variety of molecular properties for fairly large systems. Four- and twocomponents Hamiltonians are applied within the framework of density functional theory for calculations of such properties as linear molecular polarizabilities,1–5 non-linear molecular polarizabilities and other high-order optical properties,6–8 nuclear shielding constants,9–15 and indirect spin-spin coupling constants.16–20 The nuclear shielding constants and the spinspin coupling constants of heavy nuclei are of special interest in this respect, since they depend on electron density in the vicinity of the nucleus, and it has been recognized early on21 that electron velocities (and consequently the relativistic effects) are the largest in this region. The presence of a heavy nucleus in a molecule affects not only the Nuclear Magnetic Resonance (NMR) properties of the heavy nucleus in question, but influences also the shielding constants and indirect spin-spin coupling constants of the nearby light nuclei. This phenomenon is called, after Pyykkö et al.,22 the heavy-atom-on-light-atom (HALA) effect. While the relativistic effects on the shielding constants (or related to them chemical shifts) of light nuclei neighbouring heavy a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2014/140(2)/024319/8/$30.00
atoms are relatively well investigated in the literature23–30 (although mainly for proton and carbon shielding constants in halogen derivatives, much less is known about metalorganic compounds), the parallel phenomenon occurring for the nuclear spin-spin coupling constants is almost unexplored. There is a handful of papers dealing with the situation when a heavy atom mediates the geminal coupling between two light nuclei,31–35 and there seems to be a consensus that in this case the total relativistic effect is usually dominated by the scalar effects (unlike the HALA effect on the shielding constants, for which spin-orbit coupling plays a crucial role). There are, however, very few investigations concerning the situation where the heavy atom is not in the coupling path. One of those is our study on heavy metal cyanides,29 where we have shown that the relativistic effect on the 1 JCN spin-spin coupling constant may exceed 20% (for mercury cyanide) or even 40% (for gold cyanide) of the total value of the coupling. These findings inspired us to look more closely at the influence of the presence of heavy atom on the one-bond spin-spin coupling constant of the nearby light nuclei, and to investigate what factors determine the magnitude of the effect and whether the dominant role of the scalar relativistic effects is a general rule. The systems under study are derivatives of aliphatic hydrocarbons substituted with I, At, Cd, and Hg. This choice allowed us to study both one-bond carbon-carbon and onebond proton-carbon coupling constants and to explore the factors which may influence the HALA effect on these properties: the nature of the heavy atom substituent and carbon hybridization. These factors have been found important for carbon chemical shifts.30, 36 The calculations are carried out using density functional theory with the zeroth-order regular approximation (ZORA) Hamiltonian (with the spin-orbit term included), as implemented by Autschbach for the spin-spin
140, 024319-1
© 2014 AIP Publishing LLC
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coupling constants,16, 17 with the four-component DiracCoulomb Hamiltonian (as implemented recently by Saue),37 and with the non-relativistic Hamiltonian combined with scalar effective core potentials (ECPs), since our secondary aim is to compare the performance of DFT with scalar ECPs, ZORA-DFT, and Dirac-Kohn-Sham (DKS) methods for modelling of the HALA effects on the spin-spin coupling constants. The paper is organized as follows. After the description of the computational methods, we discuss the calculated 1 JCC and 1 JCH coupling constants, for each considering first the influence of the heavy atom charge and carbon hybridization on the HALA effect, and then moving to the methodological issues, comparing the results of calculations with ZORA, Dirac-Coulomb, and Schrödinger Hamiltonians, the latter both with relativistic ECPs and all-electron basis set. Finally, the results are summarized and main conclusions are presented. II. COMPUTATIONAL DETAILS A. Geometry optimization
The geometric parameters of the isolated molecules under study have been obtained by means of geometry optimization carried out using DFT with the zeroth-order regular approximation Hamiltonian38 (the spin-orbit coupling term included) as implemented in the ADF39 program with VWN40 +Becke8841 and Perdew8642 exchange-correlation functional (the functional is denoted as BP86 in ADF, but actually differs from the functional usually indicated by this acronym by using VWN instead of the local correlation PZ81 functional43 ) and the TZ2P basis set. The same geometry parameters (obtained by means of spin-orbit ZORA calculation) have been used for all (DKS, ZORA, ECP) calculations of the spin-spin coupling constants in order to separate the effects of a different computational model on the molecular geometry and the spin-spin coupling constants. B. DKS calculations of the spin-spin coupling constants
The four-component DKS calculations of the spin-spin coupling constants have been carried out with a local version of the DIRAC program,44 including the recent developments by Saue.37 The large and small components of the wave function have been connected by unrestricted kinetic balance, as implemented in DIRAC.44 The Perdew, Burke, and Ernzerhof functional with Adamo and Barone’s HF exchange contribution (PBE0)45 has been used together with the uncontracted triple-ζ basis set with additional tight functions (aug-cc-pVTZ-J46 ) for carbon and hydrogen, and uncontracted triple-ζ Dyall’s basis set (dyall.v3z47 ) for mercury, cadmium, iodine, and astatine. All calculations have been performed with the Gaussian charge distribution model, as default in DIRAC. C. ZORA calculations of the spin-spin coupling constants
The ZORA results have been obtained using the DFT as implemented in the ADF program, employing the PBE0
J. Chem. Phys. 140, 024319 (2014)
hybrid functional. (First-order potential of the hybrid PBE0 functional have been used during the calculations of the spinspin coupling constants.) The results obtained with both onecomponent scalar ZORA Hamiltonian (denoted as sc-ZORA) and with two-component spin-orbit ZORA Hamiltonian (denoted so-ZORA) will be presented. We have used triple-ζ Slater basis set with additional tight functions (jcpl48 ) for hydrogen, carbon, mercury, and iodine and the TZ2P (tripleζ +2 polarization functions basis set) for cadmium and astatine. All calculations have been performed with the Gaussian charge distribution model. D. ECP calculations of the spin-spin coupling constants
The effective core potential results have been obtained using the Gaussian 0949 program. We have employed the PBE0 functional, the aug-cc-pVTZ-J basis set for carbon and hydrogen, and several effective core potentials for the heavy elements: LANL2DZ for iodine,50 cadmium,51 and mercury,51 MWB28 for cadmium,52 MWB46 for cadmium53 and iodine,54 MWB60 for mercury,52 MWB78 for astatine55 and mercury,55 MDF28 for cadmium56 and iodine,57 MDF46 for iodine,58 MDF60 for astatine57 and mercury,59 and MDF78 for astatine.58 E. Non-relativistic calculation of the spin-spin coupling constants
Two sets of non-relativistic calculations of the spin-spin coupling constants have been performed. The first set has been carried out using DFT as implemented in the ADF program with the PBE0 functional. We have used the tripleζ Slater jcpl basis set available in the ADF program for hydrogen, carbon, mercury, and iodine and TZ2P for cadmium and astatine. The second set of calculations has been performed using DFT as implemented in Dalton 201160 program. The PBE0 functional, the Gauss-type uncontracted aug-cc-pVTZ-J basis set for carbon and hydrogen, and the uncontracted dyall.v3z basis set47 for mercury, cadmium, iodine, and astatine have been used. This allows us to estimate the effect of using different basis sets in ADF and DIRAC calculations. III. RESULTS AND DISCUSSION
Below, we are going to discuss the influence of the presence of a heavy atom on the spin-spin coupling constants between two light nuclei (first 1 JCC , then 1 JCH ). For each coupling constant, we first consider the influence of various factors (the charge of the heavy nucleus, the carbon hybridization, the relative magnitude of the scalar and spin-orbit terms), and then compare the ZORA-DFT and DKS results and discuss the performance of the ECPs in rendering the scalar relativistic effects (Secs. III A 2 and III B 2, respectively). All results discussed in Subsections III A 1 and III B 1 have been obtained with the ADF package using the ZORA or non-relativistic Hamiltonians. Additional calculations, reported in Secs. III A 2 and III B 2, have been performed with
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J. Chem. Phys. 140, 024319 (2014)
TABLE I. The 1 JCC coupling constants calculated with the PBE0 functional using different Hamiltonians. Jcpl basis sets (TZ2P for cadmium and astatine) have been used for nonrelativistic, sc-ZORA, and so-ZORA calculations, aug-cc-pVTZ-J basis set for carbon and hydrogen, and dyall.v3z basis set for cadmium, mercury, iodine, astatine for DKS calculations. The relativistic term has been calculated as difference between so-ZORA and nonrelativistic results. Nonrelativistic (Hz)
sc-ZORA (Hz)
so-ZORA (Hz)
Relativistic term (Hz)
DKS (Hz)
Substituent effect (Hz)
ICCH ICHCH2 ICH2 CH3 AtCCH AtCHCH2 AtCH2 CH3 CH3 CdCCH CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCCH CH3 HgCHCH2 CH3 HgCH2 CH3
208.9 80.2 34.0 199.3 80.0 34.6 125.2 59.2 31.0 126.0 59.1 30.6
207.7 79.7 33.6 193.5 78.9 34.0 130.2 60.8 31.4 144.1 64.9 32.1
206.3 80.1 34.1 176.2 78.9 35.6 130.3 60.7 31.3 146.0 64.9 32.0
− 2.6 − 0.1 0.1 − 23.1 − 1.1 1.0 5.1 1.5 0.3 20.0 5.9 1.4
206.8 79.7 34.6 172.7 78.2 36.2 130.2 60.6 30.8 144.7 64.7 32.3
9.0 8.1 1.8 − 21.1 6.8 3.3 − 67.0 − 11.4 − 1.0 − 51.2 − 7.1 − 0.3
HCCH CH2 CH2 CH3 CH3
197.2 72.2 32.4
197.3 72.1 32.3
197.3 72.0 32.3
... ... ...
... ... ...
... ... ...
the Dirac-Coulomb Hamiltonian and Schrödinger Hamiltonian combined with scalar relativistic ECPs. A. 1 JCC coupling constants
The 1 JCC coupling constants calculated using the Schrödinger, ZORA and Dirac-Coulomb Hamiltonians are shown in Table I. Table I contains also the “relativistic effect,” calculated as a difference between the so-ZORA and non-relativistic results, and the “substituent effect,” calculated as a difference between the so-ZORA result for the compound under study and the corresponding unsubstituted aliphatic hydrocarbon. 1. General considerations a. Influence of the charge of the heavy nucleus. The relativistic effects for the 1 JCC spin-spin coupling constants under study are sizable for substituents from the 6th row of periodic table (for mercury and astatine compounds) in comparison with the total substituent effects, but nearly negligible (10% or less) for the 5th row compounds. In the case of mercury compounds, taking into account the relativistic effects significantly lower the calculated substituent effects. The same can be observed for cadmium derivatives, but in this case the relativistic effect is only about 10% of the total substituent effect. For iodine derivatives the relativistic effect on 1 JCC is negligible (of the order of magnitude of numerical accuracy), with the exception of ICCH, where it lowers the calculated substituent effect by about 30%. In the case of AtCCH, the total substituent effect is actually dominated by the relativistic effect (the non-relativistic calculations lead to a very similar result as for HCCH). For other astatine derivatives, the relativistic effect is smaller, but still quite significant in comparison with the substituent effect. (Of the same sign for AtCH2 CH3 , of the opposite sign for AtCHCH2 .) The magnitude of the relativistic effect changes strongly with carbon hybridization (as discussed in more detail below),
but the trends in these changes are very similar in both series of structural analogs. The ratio between the relativistic effects for astatine derivative and its iodine analog is always about 11, and about 4 for mercury derivative and its cadmium analog. b. Influence of the carbon hybridization. The magnitude of the substituent and relativistic effects on the 1 JCC spin-spin coupling constants depends strongly on the hybridization of carbon atoms. The largest effects are observed for the systems with the sp hybridization and the smallest (at least one order of magnitude smaller) effects for the systems with the sp3 hybridization (except for halides, where the relativistic effects for the sp2 and sp3 hybridization are of comparable magnitude). It is however worth noting that for the compounds containing the 12th group elements the relative contribution of the relativistic effect to the substituent effect actually increases when going from the sp to sp3 hybridization, since the substituent effect decreases to a larger extent than the relativistic effect. c. Influence of the spin-orbit coupling. The ratio of the scalar and spin-orbit contributions to the total relativistic effect on the 1 JCC spin-spin coupling constant depends strongly on the nature of the heavy atom substituent. For halides both contributions are, as a rule, of similar magnitude, except for AtCHCH2 , where the scalar term dominates. Very small values of the relativistic effect on 1 JCC in ICHCH2 and ICH2 CH3 result from mutual cancelling of the scalar and spin-orbit term, each about 0.5 Hz, but with the opposite signs. In mercury and cadmium derivatives, nearly the total relativistic effect comes from the scalar term, since the spin-orbit term contributes less than 10% to it (except for CH3 CdCH2 CH3 , but here the relativistic effect is very small). Larger role of spin-orbit coupling in the molecular properties of halides seems a general rule, connected with p-character of the heavy atom.
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TABLE II. Comparison of the individual terms of 1 JCC in AtCCH and CH3 HgCCH, obtained using nonrelativistic, sc-ZORA, and so-ZORA Hamiltonians. AtCCH (Hz)
CH3 HgCCH (Hz)
Nonrelativistic (ADF) DSO PSO FC SD
199.3 0.2 10.4 176.2 12.4
126.1 0.3 4.4 111.3 10.1
sc-ZORA DSO PSO FC SD
193.5 0.2 11.1 169.7 12.5
144.2 0.3 5.8 127.5 10.6
so-ZORA DSO PSO FC SD FC+SD/PSO cross term
176.2 0.2 10.3 152.8 10.5 2.4
146.0 0.3 5.9 129.4 10.8 − 0.3
Inspired by the work of Autschbach et al.,17 we decided to examine the influence of the spin-orbit coupling on the individual terms of the spin-spin coupling constant. In the twocomponent computation, there exists a cross term between the spin-dependent Fermi contact (FC) and spin-dipole (SD) terms, and the paramagnetic spin-orbital (PSO) term, denoted later as FC+SD/PSO. In most cases the spin-orbit contribution to spin-spin coupling is dominated by this term. We have therefore compared (Table II) the individual terms of 1 JCC in AtCCH (where the spin-orbit contribution is the largest) and in CH3 HgCCH, where the spin-orbit contribution is negligible with respect to the scalar term. The results show that in the case of CH3 HgCCH the FC+SD/PSO cross term is about 17% of the spin-orbit contribution (about −0.3 Hz), and approximately 15% of the spin-orbit contribution (about 2.4 Hz) in AtCCH. Therefore, unlike for the couplings of heavy nuclei analyzed by Autschbach et al.,17 the FC+SD/PSO cross term is sizeable for 1 JCC , but does not dominate the spin-orbit contribution. Inclusion of the spin-orbit coupling changes predominantly the pure FC term. 2. Comparison of different computational methods
In this paragraph we compare the one-component and two-component ZORA results with the results of other available computational methods including the relativistic effects: four-component all-electron Dirac-Kohn-Sham method and effective core potentials. It should be stressed that comparison of ZORA and Dirac-Kohn-Sham results or ZORA and ECP results is not straightforward, since, out of necessity, different basis sets are used (ADF employs Slater orbital basis set, while in DIRAC or Gaussian program Gauss orbitals are used). For that reason we have also performed comparison of the results obtained using the Slater and Gauss orbitals with the non-relativistic Hamiltonians. Very good agreement between the results obtained with the jcpl (Slater type) basis set and the aug-cc-pVTZ-J/dyall.v3z (Gauss type) has been ob-
served (see Ref. 61 for the data in Table 1). In most cases the results differ less than 0.5 Hz (for several cases the difference is between 0.5 and 1.0 Hz). Only for AtCCH a significantly larger difference is observed (about 2.8 Hz). Considering this, the use of different basis sets should not influence materially the comparison between the ZORA-DFT and DKS results, except when the relativistic effect is very small. a. Comparison between so-ZORA and DKS. Inspection of the results in Table I leads to the conclusion that so-ZORA reproduces the DKS results very well. Explicit inclusion of small spinor during the calculation does not change significantly the calculated 1 JCC couplings in comparison with twocomponent ZORA approximation: in most cases the differences are less than 1 Hz, and seems to be caused mostly by the change of the basis set (see above) rather than the picture change effects. Only for the systems with the heaviest metallic substituents and sp-hybridized carbon nuclei (i.e., CH3 HgCCH and AtCCH) the ZORA-DFT—DKS difference is slightly bigger (e.g., 3.4 Hz in the case of AtCCH) but a significant part of this change can be again attributed to the basis set effect (about 2.8 Hz, see above). b. Comparison between sc-ZORA and ECPs methods. The JCC spin-spin coupling constants calculated using scalar ECPs are compared in Table III with the one-component scalar-only ZORA results. Again, it should be pointed out that a different basis set has been used for sc-ZORA and ECP calculations. Two types of ECPs have been investigated: Los Alamos ECPs (LANL2DZ) and series of Stuttgart ECPs (MDF and MWB with small and big core replacement). LANL2DZ does not reproduce the sc-ZORA results very well, and actually it does not offer any consistent improvement over the non-relativistic calculations. The same is true of the large-core MDF and MWB pseudopotentials, so it seems to be mainly a matter of replacing too many core electrons by the effective potential. The small-core MWB results are in excellent agreement with sc-ZORA. A slightly worse agreement is observed for small-core MDF results, but they are still a big improvement over the non-relativistic calculations, and the differences between the MWB60 and MDF60 results are usually very small. One of a very few cases when a sizeable discrepancy (about 4 Hz) between the MWB60 and MDF60 results is observed is 1 JCC in CH3 HgCCH. We have performed additional calculations with a more recent version of MDF6056 (denoted in Table III as MDF60(new)) and obtained results much closer to MWB60; thus the observed difference is most likely a consequence of different fitting procedures employed to obtain the two ECPs.
1
B. 1 JCH coupling constants 1
JCH coupling constants in the systems under study are of two types, as far as the placement of the coupled nuclei with respect to the heavy atom is concerned. First we are going to discuss the 1 JCH coupling constants of the carbon nuclei directly bound with the heavy atom (1 JCα H ), and later the 1 JCH coupling constants involving the carbon nuclei in beta
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TABLE III. Comparison of 1 JCC calculated with different types of effective core potentials on heavy atoms and aug-cc-pVTZ-J basis set on C and H, using PBE0 functional.
ICCH ICHCH2 ICH2 CH3 AtCCH AtCHCH2 AtCH2 CH3 CH3 CdCCH CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCCH CH3 HgCHCH2 CH3 HgCH2 CH3
LANL2DZ (Hz)
MDF Largea (Hz)
MDF Smallb (Hz)
MDF60 Newc (Hz)
MWB Largea (Hz)
MWB Smallb (Hz)
sc-ZORA (Hz)
210.0 79.5 33.5 ... ... ... 121.3 58.0 30.7 128.0 60.8 31.1
209.5 79.4 33.5 197.9 78.9 33.9 ... ... ... ... ... ...
207.6 79.3 33.6 194.3 78.6 34.0 131.4 60.7 31.4 140.6 63.1 31.8
... ... ... ... ... ... ... ... ... 144.1 64.5 32.1
210.7 79.5 33.5 196.5 78.7 33.9 120.4 151.8 30.7 130.6 60.9 31.3
... ... ... ... ... ... 131.5 60.8 31.5 144.3 64.6 32.1
207.7 79.7 33.6 193.5 78.9 34.0 130.2 60.8 31.4 144.2 64.9 32.2
a
Large-core pseudopotential. Small-core pseudopotential. c Small-core pseudopotential (60 core electrons replaced by a pseudopotential in Hg) reparametrized.56 b
position with respect to the heavy atom. In the latter case, for the system of the RCHCH2 type, the coupled proton can be either in cis or in trans position with respect to the heavy atom, and for the RCH2 CH3 systems the dihedral angle between the heavy atom–alpha carbon and hydrogen–beta carbon may play a role, so these factors will be discussed. The computational results are presented in Table IV. 1. General considerations a. Influence of the charge of the heavy nucleus. The relativistic effects on the 1 JCH coupling constants in iodinesubstituted hydrocarbons are 1 Hz or less (even when carbon is in the α position with respect to the heavy atom). Somewhat larger effects (but still rather smaller than those resulting from other changes in computational protocol, e.g., the use of a different basis set) can be observed for astatinesubstituted analogs. The only case when the relativistic effect on 1 JCH constitutes a majority of the substituent effect (apart from CH3 CdCH2 CH3 , where the substituent effect is very small) takes place in organomercury compounds—there the relativistic effect can even exceed 10 Hz. The ratio between the relativistic effect for 1 JCα H in organomercury compound and its cadmium analog is about 3 (similarly as for 1 JCC ), and a similar regularity can be observed for the 1 JCβ H couplings (but not for iodine/astatine analogs). Not surprisingly, the relativistic effect weakens with the distance of the coupled nuclei from the heavy atom: it is much larger for 1 JCα H than for 1 JCβ H . In the case of 1 JCβ H for cadmium compound with sp2 and sp3 carbon hybridization the relativistic contributions do not exceed 1.4 Hz, and for iodine compounds they are even smaller (about 0.4 Hz). The biggest contribution for 1 JCβ H coupling constants is observed for CH3 HgCCH, where the relativistic term is 10.0 Hz. Interestingly, the relativistic calculations lead in this case to much smaller substituent effect than the non-relativistic ones. b. Influence of the carbon hybridization and molecular geometry. Similarly as for 1 JCC , carbon hybridization influences
the magnitude of the relativistic effects on 1 JCα H . They are bigger for the systems with the sp2 hybridization than for the systems with the sp3 hybridization (compare 15.3 Hz for CH3 HgCHCH2 and 6.6 Hz for CH3 HgCH2 CH3 ). Only for iodine derivatives a reverse trend is observed but in this case the relativistic terms are small (less than 0.7% of the total spinspin coupling constant) and therefore sensitive to numerical noise. In the case of 12th group of periodic table and astatine compounds the ratio between the relativistic effects for the analogous systems with sp2 and sp3 hybridization is about 2–2.5. The relativistic effects on 1 JCβ H are visibly affected by the heavy atom–carbon–carbon-hydrogen dihedral angle. When cis and trans configurations of cadmium and mercury derivatives are compared, always the coupling of the proton in the trans position is more affected by the relativistic effects (and exhibits larger variation with carbon hybridization). This is not necessarily true for iodine and astatine derivatives, but, as mentioned above, the relativistic effects are usually negligibly small there. c. Influence of the spin-orbit coupling. A comparison of the so-ZORA and sc-ZORA results shows that the spin-orbit coupling effect is as a rule negligible in comparison with the scalar term for all types of 1 JCH , except for astatine derivatives. For 1 JCα H in AtCHCH2 the spin-orbit coupling term is about −4.9 Hz (the scalar term is only 0.4 Hz), while for 1 JCβ H in AtCCH it is about −4.7 Hz (to be compared with the scalar term of 0.2 Hz) and for 1 JCβ Htrans in AtCHCH2 it is about −2.0 Hz (the scalar term 0.1 Hz). It is noticeable that for astatine derivatives with sp3 hybridization the spin-orbit coupling term is much smaller than for derivatives with sp2 and sp carbon.
2. Comparison of different computational approaches a. Comparison between so-ZORA and DKS. Inspection of the results (see Table IV) for all 1 JCH confirms that the two-component ZORA approximation reproduces the
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TABLE IV. 1 JCH calculated with PBE0 functional at different levels of theory. The Jcpl basis sets (TZ2P for cadmium and astatine) have been used for nonrelativistic, sc-ZORA, and so-ZORA calculations, aug-cc-pVTZ-J basis set for carbon and hydrogen, and dyall.v3z basis set for cadmium, mercury, iodine, astatine for DKS calculations. The relativistic term has been calculated as a difference between so-ZORA and nonrelativistic results.
1J Cα H
1J Cβ Hcis
1J Cβ Htrans
1J Cβ H
1J Cα H 1J CH
–CH3
ICHCH2 ICH2 CH3 AtCHCH2 AtCH2 CH3 CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCHCH2 CH3 HgCH2 CH3 ICHCH2 ICH2 CH3 AtCHCH2 AtCH2 CH3 CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCHCH2 CH3 HgCH2 CH3 ICHCH2 ICH2 CH3 AtCHCH2 AtCH2 CH3 CH3 CdCHCH2 CH3 CdCH2 CH3 CH3 HgCHCH2 CH3 HgCH2 CH3 ICCH AtCCH CH3 CdCCH CH3 HgCCH CH3 CdCCH CH3 HgCCH CHCH CH2 CH2 CH3 CH3 CH4
Nonrelativistic (Hz)
sc-ZORA (Hz)
so-ZORA (Hz)
Relativistic term (Hz)
DKS (Hz)
Substituent effect (Hz)
196.1 151.9 197.0 153.8 139.0 126.0 137.5 125.0 167.9 131.6 167.7 131.6 159.0 124.7 160.9 125.3 161.7 122.2 162.1 122.4 153.3 124.1 151.2 122.9 270.7 269.6 239.7 237.4 129.7 128.1 262.2 159.1 125.2 125.7
197.2 152.5 197.4 154.5 143.8 128.3 152.1 131.6 167.7 131.5 167.2 131.2 159.1 125.0 160.3 125.8 162.4 122.6 162.2 122.4 154.6 124.5 155.8 124.4 271.6 269.4 242.6 246.8 132.1 135.0 262.4 159.2 125.3 125.8
197.0 153.0 192.5 155.2 143.7 128.2 152.8 131.6 167.6 131.4 166.8 130.7 159.1 124.9 160.3 125.8 161.9 122.6 160.2 122.4 154.6 124.5 156.1 124.3 271.1 264.7 242.6 247.4 132.0 134.9 262.4 159.1 125.2 125.8
0.9 1.1 − 4.5 1.3 4.7 2.2 15.3 6.6 − 0.3 − 0.2 − 0.9 − 0.9 0.0 0.2 − 0.6 0.5 0.2 0.4 − 1.9 0.0 1.4 0.4 4.9 1.5 0.4 − 4.9 2.9 10.0 2.3 6.8 ... ... ... ...
195.9 152.9 190.9 155.0 143.4 127.8 151.9 131.0 166.8 131.1 166.0 130.3 158.2 124.7 159.3 125.4 160.9 122.2 159.1 121.9 153.9 124.3 155.7 124.4 270.1 263.7 241.6 247.1 131.3 133.9 ... ... ... ...
37.9 27.8 33.4 30.0 − 15.4 2.9 − 6.3 6.4 8.5 6.2 7.7 5.4 − 0.1 − 0.3 1.2 0.6 2.8 − 2.6 1.0 − 2.8 − 4.5 − 0.8 − 3.0 − 0.9 8.7 2.3 − 19.8 − 14.9 6.2 9.1 ... ... ... ...
four-component DKS results very well, especially when the fact that two different basis sets (Slater- and Gauss-type) are used is taken into account. The biggest difference between soZORA/DFT and DKS (about 1.6 Hz) is observed for 1 JCα H in AtCHCH2 , but it seems to originate from different basis sets used (compare the non-relativistic results obtained with the Slater- and Gauss-type basis sets tabulated in Table 2 in the supplementary material61 ). For the other 1 JCH spin-spin coupling constants the differences between so-ZORA/DFT and DKS are in most cases less than 1.0 Hz and also seem to come mostly from using different basis sets. Thus, we conclude that so-ZORA/DFT is very efficient in rendering HALA effects on the 1 JCH coupling constants. In fact, in most cases (apart from astatine derivatives), using one-component scalar ZORA is sufficient.
b. Comparison between sc-ZORA and ECPs methods. The JCH coupling constants calculated using different scalar relativistic effective core potentials are compared with the sc-
1
ZORA results in Table 3 in the supplementary material.61 The observations are similar as for the 1 JCC coupling constants. ECPs in general seem suitable for the purpose, and the best results are obtained when small-core ECPs are employed. Among small-core ECPs, MWB appears to perform slightly better than MDF, but the differences are small.
IV. SUMMARY AND CONCLUSIONS
The 1 JCC and 1 JCH spin-spin coupling constants have been calculated by means of density functional theory for a set of derivatives of aliphatic hydrocarbons substituted with I, At, Cd, and Hg in order to evaluate the substituent and relativistic effects for these properties. The methods applied range, in order of reduced complexity, from Dirac-KohnSham method (density functional theory with four-component Dirac-Coulomb Hamiltonian), through DFT with two- and one-component zeroth order regular approximation Hamiltonians, to scalar non-relativistic effective core potentials with
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the non-relativistic Hamiltonian. The most important observations can be summarized as follows. The relativistic effects on the spin-spin coupling constants under study are at most about 25 Hz (for 1 JCC in organomercury compounds), and constitute up to 30% of the substituent effect (defined as a difference between the value of the coupling in the compound under study and the corresponding hydrocarbon). Predictably, they vary with the charge of the heavy nucleus. For 1 JCC , the ratio between relativistic effects for astatine derivative and its iodine analog is always about 11, and for mercury derivative and its cadmium analog about 4. The ratio between the relativistic effect for 1 JCα H in organomercury compound and its cadmium analog is about 3 and a similar regularity can be observed for the 1 JCβ H couplings (but not for the iodine/astatine analogs). The magnitude of the relativistic effects on the 1 JCC and 1 JCH coupling constants depends very much on carbon hybridization. For both couplings the largest effects are observed for the sp hybridization. As expected, the relativistic effects are larger for 1 JCα H than for 1 JCβ H , although even for the latter they can be non-negligible (10 Hz for CH3 HgCCH). 1 JCβ H are larger when the proton is in the trans position to the heavy substituent than when it is in a cis position. A comparison of the results obtained by means of different methods of including the relativistic effects indicates that ZORA-DFT (with the spin-orbit term included) reproduces the DKS results very well. The scalar contributions dominate the total relativistic effect on 1 JCC and 1 JCH in mercury and cadmium derivatives, while for 1 JCC and 1 JCH in halides scalar and spin-orbit effects contribute to a similar degree. The performance of scalar ECPs depends, obviously, on the relative weight of the scalar and relativistic effects, but when the former are dominant, ECPs reproduce correctly the results obtained with more elaborate relativistic method, provided the outer core electrons on the heavy atom are accounted for explicitly (“small core” types of ECPs).
ACKNOWLEDGMENTS
This work has received support from the Polish National Science Centre via the Grant No. N N204 148565, and from the Wrocław Centre for Networking and Supercomputing through a grant of computer time. MPD/2010/4 project, realized within the MPD programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, is acknowledged for a fellowship to A. Wody´nski. The project has been carried out with the use of CePT infrastructure financed by the European Union - the European Regional Development Fund within the Operational Programme “Innovative economy” for 2007-2013. 1 P.
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